Fully extended $\boldsymbol{r}$-spin TQFTs

We prove the $r$-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer $r$: The 2-groupoid of 2-dimensional fully extended $r$-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced $\textrm{Spin}_2^r$-action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the $r$-th power of their Serre automorphisms. For $r=1$ we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to $r=2$. To construct examples, we explicitly describe $\textrm{Spin}_2^r$-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau--Ginzburg models gives rise to fully extended spin TQFTs, and that half of these do not factor through the oriented bordism 2-category.

We prove the r-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer r: The 2-groupoid of 2dimensional fully extended r-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced Spin r 2 -action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the r-th power of their Serre automorphisms. For r = 1 we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to r = 2.
To construct examples, we explicitly describe Spin r 2 -homotopy fixed points in the equivariant completion of any symmetric monoidal 2category. We also show that every object in a 2-category of Landau-Ginzburg models gives rise to fully extended spin TQFTs, and that half of these do not factor through the oriented bordism 2-category.

Introduction and summary
The spin group Spin n in dimension n is by definition the double cover of the group of rotations SO n in Euclidean space R n . A spin structure on an n-dimensional oriented manifold is a lift of its tangent bundle along the covering Spin n −→ SO n . Such geometric structures and their close cousins in Lorentzian geometry are fundamental in theoretical physics, since e. g. electrons are classically modelled as sections of spin bundles.
More generally, for any continuous group homomorphism ξ : G −→ GL n , a tangential structure on an n-dimensional manifold M is a principal G-bundle on M together with a bundle map to the frame bundle of M that is compatible with ξ (see Section 2.1 for details). The case of spin structures is precisely when ξ is the covering map Spin n −→ SO n post-composed with the inclusion SO n ⊂ GL n ; in the case of orientations ξ is just that inclusion, while in the case of framings ξ is the inclusion of the trivial group into GL n .
Given the relevance of spin structures in physics, and the motivation to study functorial topological quantum field theories (TQFTs) as a means to gain insight into physics, it is natural to consider spin TQFTs. These are (higher) symmetric monoidal functors on (higher) categories of bordisms with prescribed spin structures. The case of closed spin TQFTs in dimension n = 2 was first considered in [MS, BT, NR, RS], and in [MS, StSz] they were classified 1 in terms of "closed Λ 2 -Frobenius algebras" (see Section 2.2 for the definition). Such algebraic structures formalise the relation between topological Neveu-Schwarz and Ramond sectors, examples of which can be obtained as a Z 2 -graded version of the centre construction of [LP]. In particular, there is a (1|1)-dimensional example in Vect Z 2 C whose associated TQFT computes the Arf invariant of spin surfaces. Not many other explicit examples have been studied in the literature, and all previously known classes of examples are constructed from semisimple algebraic data.
In the setting of symmetric monoidal (∞, n)-categories, fully extended TQFTs with G-structure are widely believed to be classified by homotopy fixed points of a G-action (induced from the G-action on framed bordisms) on the maximal ∞-subgroupoids of fully dualisable objects in the target (∞, n)-categories. This is described in significant, yet non-exhaustive, detail in [Lu]. To our knowledge, this general version of the cobordism hypothesis, originally put forward in [BD], is established as a theorem only up to a completion of the extended proof sketch in [Lu], or up to a conjecture on the relation between factorisation homology and adjoints, see [AF,Conj. 1.2].
On the other hand, in dimension n = 2 and in the setting of (weak) 2-categories the cobordism hypothesis for the framed and oriented case was proved explicitly in [Ps] and [HSV, HV, He], respectively: For any symmetric monoidal 2-category B the 2-groupoid of fully extended framed TQFTs Bord fr 2,1,0 −→ B is equivalent to the maximal sub-2-groupoid (B fd ) × of fully dualisable objects in B, while fully extended oriented TQFTs Bord or 2,1,0 −→ B are described by SO 2 -homotopy fixed points. The latter are objects of a 2-groupoid [(B fd ) × ] SO 2 and correspond to pairs (α, λ), where α ∈ B fd and λ : S α ∼ = 1 α is a trivialisation of the Serre automorphism of α. In Sections 3.1.4 and 3.3.1-3.3.3 we recall the notions just mentioned, in particular how the Serre automorphism S α : α −→ α, defined in (3.23), corresponds to one full rotation of frames.

r-spin cobordism hypothesis
In the present paper we classify fully extended spin TQFTs valued in an arbitrary symmetric monoidal 2-category B (Section 3), and we construct a number of examples (Section 4). More precisely, we consider r-spin TQFTs for any positive integer r. Recall that while for n 3, the double cover Spin n −→ SO n is also the universal cover, this is not true for n 2. Hence there is less reason to single out double covers of SO 2 and instead consider the r-fold cover Spin r 2 −→ SO 2 for all r ∈ Z 1 . 2 Note that necessarily Spin r 2 ∼ = SO 2 as groups, and that by definition Spin 2 = Spin 2 2 and Spin 1 2 = SO 2 . Following [SP], in Section 3.2 we describe a 2-category Bord r-spin 2,1,0 of bordisms with r-spin structure related to ξ : Spin r 2 −→ SO 2 ֒−→ GL 2 , and in Section 3.3.3 we construct a 2-category 2D r (B fd ) whose objects are pairs (α, θ), where α ∈ B fd and θ : S r α ∼ = 1 α . Then we prove (Lemma 3.18 and Theorem 3.19): Theorem (r-spin cobordism hypothesis). Let B be a symmetric monoidal 2-category and let r ∈ Z 1 . The 2-groupoid of fully extended r-spin TQFTs valued in B is equivalent to the homotopy fixed points [(B fd ) × ] Spin r 2 . This in turn is equivalent to 2D r ((B fd ) × ), and under these equivalences we have Put differently, (fully) extended r-spin TQFTs are classified by what they assign to the positively framed point + ∈ Bord r-spin 2,1,0 together with a trivialisation of the r-th power of the associated Serre automorphism. The main ingredients of the proof are a generators-and-relations presentation of Bord r-spin 2,1,0 , inspired by the work [HV], and an explicit description of r-spin bordisms in terms of holonomies, following [RW].

Examples
The choice of target 2-category is essential for extended TQFTs. To broaden the class of known r-spin TQFTs, in Section 4 we explicitly describe Spin r 2 -homotopy fixed points in the "equivariant completion" B eq of any given symmetric monoidal 2-category B. As introduced in [CR] and reviewed in Section 4.1, objects in B eq are pairs (α, A), where α ∈ B and A ∈ B(α, α) is endowed with the structure of a ∆-separable Frobenius algebra, while 1-and 2-morphisms are bimodules and bimodule maps. We show (see Corollary 4.9,and (4.5) for the definition of the Nakayama automorphism γ A : A −→ A): Proposition. Let (α, A) ∈ B eq be such that α ∈ B fd as well as S r α ∼ = 1 α and γ r A = 1 A in B. Then there is an r-spin TQFT Z : Bord r-spin 2,1,0 −→ B eq + −→ (α, A) .
( 1.2) Moreover, in Section 4.1.5 we explain how to compute the invariants such TQFTs associate to r-spin surfaces, by explicitly constructing the closed Λ r -Frobenius algebras which classify the underlying non-extended TQFTs.
An advantage of considering B eq -valued (as opposed to B-valued, for a given B) TQFTs is as follows. As explained in Remark 3.28, r-spin TQFTs valued in a pivotal 2-category B cannot detect all r-spin structures if r 3. However, the equivariant completion B eq of a pivotal 2-category B is itself not pivotal.
As a specific example of a target B, in Section 4.2 we consider the symmetric monoidal 2-category LG of Landau-Ginzburg models, constructed in [CM1,CMM]. (Examples of extended 2-spin TQFTs were first considered in [Gun].) Objects of LG are "potentials" W ∈ k[x 1 , . . . , x n ] that describe isolated singularities, and Hom categories are homotopy categories of matrix factorisations. In [CMM] it was observed that every object in LG is fully dualisable, and that precisely those potentials W (x 1 , . . . , x n ) that depend on an even number of variables give rise to fully extended oriented TQFTs. Moreover, these oriented TQFTs indeed extend the closed TQFTs associated to the (generically non-semisimple) Jacobi algebras Jac W to the point. In light of the r-spin cobordism hypothesis proved in Section 3, it is straightforward to extend these results as follows (Theorem 4.17): 3

Theorem.
Every object W (x 1 , . . . , x n ) ∈ LG gives rise to an extended 2spin TQFT valued in LG. These TQFTs factor through the oriented bordism 2-category iff n is even.
Explicitly, the 2-spin TQFT associated to an object W ∈ LG with an odd number of variables consists of the even Neveu-Schwarz sector Jac W ∈ Vect k ⊂ Vect Z 2 k and the odd Ramond sector Jac W [1] ∈ Vect Z 2 k , together with the structure maps described in general in Section 3.1.5. Moreover, in Example 4.18 we illustrate how to apply our results on equivariant completion (Section 4.1) to a variant of B = LG and explicitly compute the invariants of r-spin tori in the simplest non-trivial (and novel) example.

Examples not treated in this paper
We close this introductory section with a few comments on potential further applications of the r-spin cobordism hypothesis. Besides the 2-categories Alg k and LG (as well as their variants with additional Z 2 -, Z-or Q-gradings), it is natural to consider the 2-category Var of [CW] of smooth projective varieties and derived categories (see also Example 3.3), which appears in the study of B-twisted sigma models. The 2-category Var has a natural symmetric monoidal structure [Ba]. As explained in [Lu, CW], the Serre automorphism S X of X ∈ Var can be identified with the Serre functor of the derived category associated to X.
In [Ku], Kuznetsov constructs "fractional Calabi-Yau categories" A X as the admissible subcategories of semiorthogonal decompositions of derived categories of certain varieties X ∈ Var. This means in particular that A X is a triangulated category with suspension functor Σ, such that A X has a Serre functor S which satisfies S q ∼ = Σ p for some p, q ∈ Z with q = 0. It follows that the orbit category A X /Z has a Serre functor whose (p − q)-th power is trivialisable, see e. g. [Gr,Thm. 5.14].
It is tempting to expect that some of the fractional Calabi-Yau categories constructed in [Ku,Sect. 4] classify (p − q)-spin TQFTs whose target is Var up to the Z-action quotiented out in orbit categories. This is possible only if one can identify the Serre functor of A X /Z with the Serre automorphism of some other object in the target 2-category. More generally, we could work in the 2-category of smooth and proper triangulated differential graded categories described in [BFK, App. A]. In this setting, both the geometric constructions of [Ku] and the representation theoretic examples of fractional Calabi-Yau categories in [Gr,Sect. 6] may lead to interesting r-spin TQFTs.

Non-extended r-spin TQFTs
In this section we review the classification of non-extended closed r-spin and framed TQFTs following [StSz], to which we refer for details. We recall the relevant categories of 2-dimensional bordisms as well as the notion of a closed Λ r -Frobenius algebra, and we state the main classification result: 2-dimensional r-spin and framed (r = 0) TQFTs are equivalent to closed Λ r -Frobenius algebras in the target category.

Framed and r-spin TQFTs
By a surface we mean a 2-dimensional compact smooth manifold. Let G be a topological group, let ξ : G −→ GL 2 (2.1) be a continuous group homomorphism, and recall that the frame bundle F Σ −→ Σ of a surface Σ is a principal GL 2 -bundle. A G-structure (more precisely: a tangential structure for ξ : G −→ GL 2 ) on Σ is a principal G-bundle π : P −→ Σ together with a bundle map q intertwining the group actions via ξ: A map of surfaces with G-structure is a bundle map which is a local diffeomorphism of the underlying surfaces. Such a map is called a diffeomorphism if its underlying map of surfaces is a diffeomorphism, and an isomorphism of G-structures if the underlying map of surfaces is the identity. We will consider the following tangential structures: • A framing is a tangential structure for the trivial group: • An orientation is a tangential structure for the inclusion where GL + 2 is the subgroup of elements in GL 2 with positive determinant.
• For r ∈ Z 0 , an r-spin structure is a tangential structure where p r : GL + 2 r −→ GL + 2 is the r-fold covering for r > 0, while for r = 0 it is the universal cover.
By a trivial r-spin structure on a surface Σ we mean an r-spin structure isomorphic to the r-spin structure with trivial bundles P = Σ× GL + 2 r , F Σ = Σ×GL 2 and trivial bundle map q (+) = id Σ ×p r (positive orientation) or q (−) = id Σ ×(T • p r ) (negative orientation), where T is composition with the matrix ( +1 0 0 −1 ) ∈ GL 2 . Remark 2.1. A 1-spin structure is the same as an orientation, and a 2-spin structure is usually called a spin structure. Moreover, we can identify framings with 0-spin structures by noting that the fibres of a 0-spin bundle are contractible, see [RS,Prop. 2.2]. This is consistent with the fact that for any r ∈ Z 0 , an r-spin structure is a Z r -bundle over the oriented frame bundle.
Let r ∈ Z 0 . There is a symmetric monoidal category of 2-dimensional r-spin bordisms Bord r-spin 2,1 as follows. An object S is a 1-dimensional closed manifold s embedded in a cylinder s × (−1, 1), together with an r-spin structure on the cylinder. For an object S we write S (+) := s × [0, 1) and S (−) := s × (−1, 0] with the restricted r-spin structures. The morphisms of Bord r-spin 2,1 are diffeomorphism classes of r-spin bordisms: For S, S ′ ∈ Bord r-spin 2,1 , an r-spin bordism S −→ S ′ is a compact surface Σ with r-spin structure, together with a boundary parametrisation map S (+) ⊔ S ′ (−) ֒−→ Σ, i. e. a map of r-spin surfaces that identifies the boundary of Σ with the 1-dimensional embedded manifolds s × {0} ⊂ S and s ′ × {0} ⊂ S ′ . Finally, a diffeomorphism of r-spin bordisms is a diffeomorphism of r-spin surfaces compatible with the boundary parametrisations. We usually refer to a morphism in Bord r-spin 2,1 by a bordism that represents it. A particular class of r-spin bordisms are deck transformation bordisms. These are cylinders whose boundary parametrisations are given by deck transformations of the r-spin bundle on the source or target object.
The composition of morphisms in Bord r-spin 2,1 is given by glueing along boundary parametrisations, hence the unit morphisms are given by cylinders with trivial boundary parametrisations. Taking disjoint unions endows Bord r-spin The case of 2-dimensional closed spin TQFTs (r = 2) was first described and classified in [MS], including concrete examples in terms of Clifford algebras viewed as objects in the category of super vector spaces C = Vect Z 2 C . Spin TQFTs were further discussed from the perspective of extended TQFTs in [Gun], and spin state sum constructions were given in [BT, NR]. TQFTs with r-spin structure for arbitrary r were introduced in [No] and further studied in [RS]. The classification of general r-spin TQFTs appears in [StSz], in terms of the algebraic structures we review next.

Classification in terms of closed Λ r -Frobenius algebras
A closed Λ r -Frobenius algebra C in a symmetric monoidal category C consists of a collection of objects C a ∈ C for all a ∈ Z r as well as morphisms for all a, b ∈ Z r . The Nakayama automorphisms of C are for all a ∈ Z r . These data by definition satisfy the following conditions: (co)associativity: (2.13) commutativity: (2.14) twist relations: A map of closed Λ r -Frobenius algebras ϕ : C −→ D is a collection of morphisms ϕ a : C a −→ D a preserving the structure morphisms. Analogously to the case of ordinary Frobenius algebras, maps of closed Λ r -Frobenius algebras are always isomorphisms.
Example 2.3. (i) One class of closed Λ r -Frobenius algebras in a given symmetric monoidal category C can be constructed from ordinary Frobenius algebras A in C whose ordinary Nakayama automorphism γ A satisfies γ r A = 1 A (see Section 4.1.1 and (4.5) below for details). Indeed, as explained in [RS] and [StSz,Sect. 4.2], the construction of commutative Frobenius algebras as the centres of certain types of non-commutative Frobenius algebras in [LP,Sect. 2.7] is naturally the special case of r = 1 of a construction of "Z r -graded centre" for any r ∈ Z 0 .
(ii) In the category Bord r-spin 2,1 , r-spin circles, pair-of-pants, cups, and caps naturally assemble into a closed Λ r -Frobenius algebra C. The precise presentation is given in [StSz,Sect. 5.1] in terms of a combinatorial description of r-spin structures. In particular, it follows from [StSz,Eq. (5.2)] that the Nakayama automorphisms of C are deck transformation bordisms.
The closed Λ r -Frobenius algebra C of Example 2.3(ii) is not just any example. As proven in [StSz,Thm. 5.2.1], Bord r-spin 2,1 is generated as a symmetric monoidal category by the data of C, subject to relations given by the defining properties (2.11)-(2.16). This implies: Theorem 2.4 ( [StSz,Cor. 5.2.2]). There is an equivalence of symmetric monoidal groupoids between the groupoid of r-spin TQFTs with target C and the groupoid of closed Λ r -Frobenius algebras in C.
For this reason we will refer to the objects C a of a closed Λ r -Frobenius algebra in any given symmetric monoidal category C (not necessarily equivalent to Bord r-spin 2,1 ) as the a-th circle spaces. The a-th circle space in Bord r-spin 2,1 is simply the circle with "framing number" a, and we denote it S 1 a . Below in Sections 3.1.5, 4.1.5 and 4.2 we will use Theorem 2.4 to construct examples of closed r-spin TQFTs beyond those mentioned in Section 2.1.

Computing invariants
The above classification theorem provides a way to compute invariants of rspin surfaces from r-spin TQFTs in terms of the algebraic data of a closed Λ r -Frobenius algebra. As the number of diffeomorphism classes of r-spin structures on a connected oriented surface of genus g 2 is, if non-zero, either one (r odd) or two (r even), we are mainly interested in surfaces of genus 0 and 1; see e. g.
[Sz, Sect. 3] and the references therein for a detailed account.
The sphere S 2 admits an r-spin structure only if r ∈ {1, 2}, in which case it is unique up to isomorphism, hence the torus T 2 is of most interest. Any torus with r-spin structure can be presented in terms of the closed Λ r -Frobenius algebra in Bord r-spin for some a, b ∈ Z r . Moreover, as shown in [StSz,Prop. 4.1.4], the r-spin torus T (a, b) is diffeomorphic to the r-spin torus T (gcd(a, b, r), 0), and in fact diffeomorphism classes of r-spin tori are in bijection with divisors of r. Hence we write for the class of r-spin tori corresponding to the divisor d.
Proposition 2.5 ( [StSz,Prop. 4.1.4]). The invariant of the r-spin torus T (d) computed by a C-valued closed r-spin TQFT Z classified by a closed Λ r -Frobenius algebra C is the quantum dimension of C d , where b is the braiding of C.

Fully extended r-spin TQFTs
In this section we describe fully extended r-spin TQFTs and prove the corresponding cobordism hypothesis in the 2-categorical setting. In Section 3.1 we recall some aspects of symmetric monoidal 2-categories B, their Serre automorphisms, and we construct canonical closed Λ 0 -Frobenius algebras. The short Section 3.2 describes the 2-category Bord r-spin 2,1,0 of r-spin bordisms. Then in Section 3.3 we define the 2-groupoid of fully extended r-spin TQFTs Bord r-spin 2,1,0 −→ B and explain that it is equivalent to the 2-groupoid of Spin r 2 -homotopy fixed points.

Dualisability in symmetric monoidal 2-categories
In this section we present our notational conventions for dualisability in symmetric monoidal 2-categories. Moreover, we construct a closed Λ 0 -Frobenius algebra (in the sense of Section 2.2) for every fully dualisable object. For complete definitions we refer to [Be, SP, Ps] and references therein; with an eye towards examples in Section 4, below we mostly use the same conventions as in [CMM, Sect. 2].

Conventions for 2-categories
By a 2-category we mean a (possibly non-strict) bicategory B in the sense of [SP,App. A.1]. For objects α, β ∈ B, we denote the category of 1-morphisms α −→ β by B(α, β); for 1-morphisms X, Y ∈ B(α, β), we write Hom B (X, Y ), or simply Hom(X, Y ), for the set of 2-morphisms X −→ Y . Horizontal and vertical composition are denoted by ⊗ and •, respectively: We read string diagrams from right to left and from bottom to top. For instance, for 1-morphisms X ∈ B(α, β), X ′ ∈ B(β, γ) and V ∈ B(α, γ), a 2-morphism ϕ ∈ Hom(X ′ ⊗ X, V ) is represented by though sometimes we will suppress object labels in string diagrams, as e. g. in (3.4) below.

Adjoints
Let B be a 2-category. A 1-morphism X ∈ B(α, β) has a left adjoint if there exists a 1-morphism † X ∈ B(β, α) together with adjunction 2-morphisms such that the Zorro moves are satisfied. Similarly, a right adjoint for X consists of X † ∈ B(β, α) with that satisfy analogous Zorro moves.
If X, Y ∈ B(α, β) have left and right adjoints (with chosen adjunction maps), then we write † ϕ : for the left and right adjoint of ϕ ∈ Hom(X, Y ), respectively. We call B pivotal if every 1-morphism X comes with chosen left and right adjunction data such that † X = X † , † ϕ = ϕ † for all 2-morphisms ϕ, and for all composable 1-morphisms X, Y .

Symmetric monoidal structure
Let B be a 2-category. A monoidal structure on B consists of a 2-functor : B × B −→ B (3.9) called monoidal product, a unit object (3.10) a pseudonatural transformation a : • ( × Id B ) −→ • (Id B × ) called associator, a weak inverse a − for a, as well as unitors, 2-unitors and a pentagonator (which we will usually suppress), subject to the coherence axioms in [SP,Sect. 2.3].
Viewing a monoidal 2-category B as a 3-category with a single object and using the strictification results of [Gur, Gut], we can use the 3-dimensional graphical calculus of [BMS, Tr]. For this we extend our diagrammatic conventions by reading 3-dimensional diagrams from front to back. For instance, for 1-morphisms Let B be a monoidal 2-category. Writing τ : B × B −→ B × B for the strict 2functor that acts as (ζ, ξ) −→ (ξ, ζ) on objects, 1-and 2-morphisms, a symmetric braided structure on B consists of a pseudonatural transformation for all X ∈ B(α, α ′ ) and Y ∈ B(β, β ′ ). Graphically, the 2-morphism components are depicted as (3.15)

Duality and Serre automorphism
Let B be a symmetric monoidal 2-category. An object α ∈ B has a dual if there exists an object α # together with adjunction 1-morphisms More precisely, these data witness α # as the right dual of α. Using the symmetric braiding of B, the object α # is also the left dual of α, with adjunction maps (3.21) For another 1-morphism Y ∈ B(α, β) and a 2-morphism ϕ ∈ Hom(X, Y ), its dual (3.22) An object α in a symmetric monoidal 2-category B is called fully dualisable if it has a dual α # such that the 1-morphisms ev α , coev α have both left and right adjoints (as in Section 3.1.2). The full sub-2-category of fully dualisable objects is denoted B fd , and we call B fully dualisable if B ∼ = B fd .
Convention 3.1. Whether or not a symmetric monoidal 2-category B is fully dualisable is a property of B. If it is fully dualisable, we will assume that we have chosen explicit duality data (α # , ev α , coev α ) and adjunc- Put differently, we then view B as "fully dualised".
As shown in [Ps], the adjunction 1-morphisms ev α , coev α of a fully dualisable object α do not only have left and right adjoints, but these again have left and right adjoints, and so on infinitely. The relations between multiple adjoints are negotiated by the Serre automorphism The general result [Ps,Thm. 3.9] on multiple adjoints in B fd implies in particular which together with the components S α assemble into a pseudonatural transformation Id (B fd ) × −→ Id (B fd ) × . This can be slightly generalised: Proposition 3.2. Let B be a symmetric monoidal pivotal 2-category such that B fd has adjoints for all 1-morphisms. Then the Serre automorphisms S α together with the 2-morphisms (expressed in terms of the graphical calculus of [BMS] for symmetric monoidal pivotal 2-categories) Substituting this into the proof of [HV, Prop. 2.8], we find that specifying S X amounts to filling the diagram (3.28) This is precisely what the expression of S X in (3.27) does.
Example 3.3. We sketch a few fully dualisable symmetric monoidal 2-categories that appear in connection with 2-dimensional TQFT: (i) There is a 2-category Bord fr 2,1,0 of 2-framed points, 1-dimensional bordisms and 2-dimensional bordism classes that we review below in Section 3.2. The Serre automorphism S + of the positively framed point + ∈ Bord fr 2,1,0 generates an action of π 1 (SO 2 ) ∼ = Z and corresponds to a twist of the interval over the point +.
(ii) State sum models: For k a field, there is a 2-category Alg fd k of separable kalgebras, bimodules and bimodule maps [Lu, SP]. The Serre automorphism of A ∈ Alg fd k is the A-A-bimodule Hom k (A, k).
(iii) Landau-Ginzburg models: There is a 2-category LG of isolated singularities, matrix factorisations and their maps up to homotopy [CM1,CMM], which we briefly review in Section 4.2 below. The Serre automorphism of W ∈ LG is isomorphic to 1 W up to a shift.
(iv) B-twisted sigma models: There is a 2-category Var of smooth projective varieties, Fourier-Mukai kernels and Ext groups [CW, Ba]. The Serre automorphism of X ∈ Var is given by tensoring with the canonical line bundle of X shifted by − dim(X).
(v) Topologically twisted models: There is a 2-category DGSat k of essentially small, smooth, proper and triangulated differential graded k-categories and their derived categories of bimodules [BFK, App. A]. The Serre automorphism of C ∈ DGSat k is given in terms of the k-linear dual composed with the canonical trace functor associated to C. The 2-categories of parts (iii) and (iv) are equivalent to sub-2-categories of DGSat k .

A Frobenius algebra
Let B be a symmetric monoidal 2-category. For a fixed fully dualisable object α ∈ B fd with α ## = α, we now consider the a-th circle spaces where the isomorphism in (3.29) is induced by , then for an algebra A ∈ Alg fd k the zeroth and first circle spaces are the zeroth Hochschild homology and cohomology of A, respectively: LG of Example 3.3(iii), then the circle spaces of a given object are (shifts of) the associated Jacobi algebra, as we will explain in Section 4.2 below. In the following we will sometimes treat the isomorphism in (3.29) as an identity, and we usually drop the index "α" in C α a .
Next we set for all a, b ∈ Z, where in the expressions for ε −1 and ∆ a,b we use the isomorphisms obtained from (3.25). The above data have a familiar structure (recall the definition of closed Λ 0 -Frobenius algebras in Section 2.2): Proposition 3.4. The data {C a } a∈Z and η 1 , ε −1 , {µ a,b , ∆ a,b } a,b∈Z have the properties of a closed Λ 0 -Frobenius algebra in the symmetric monoidal category b∈Z satisfy the (co)associativity, (co)unitality and Frobenius conditions is straightforward to check using the diagrammatic calculus for monoidal 2-categories. The remaining defining relations (2.14)-(2.16) of a closed Λ 0 -Frobenius algebra are more difficult to verify directly. Instead, we use the framed cobordism hypothesis (Theorem 3.8 below) to argue indirectly: To α ∈ B fd corresponds a symmetric monoidal functor Z : Bord fr 2,1,0 −→ B with Z(+) = α, such that the data η 1 , ε −1 , {µ a,b , ∆ a,b } a,b∈Z are the images under Z of 2-morphisms in Bord fr 2,1,0 . The latter in turn are generators of the framed bordism 1-category and satisfy all the relations of a closed Λ 0 -Frobenius algebras, which follows from the special case r = 0 of [StSz,Thm. 5.2.1]. Hence also their images Together with the r-spin cobordism hypothesis proved in Section 3.3 below this implies: Proof. Combine [StSz,Thm. 5.2.1] for r ∈ Z 1 with Theorem 3.19 below. This in particular guarantees the existence of isomorphisms C a ∼ = C a+r for all a ∈ Z.
3.2 The 2-category of r-spin bordisms 3.2.1 2-categories of bordisms with tangential structure Here we briefly recall 2-categories of bordisms with G-structure. For more background and details we refer to [SP,].
We begin by fixing conventions for double categories. A double category D consists of a category of objects D 0 , a category of horizontal morphisms D 1 , unit horizontal morphisms, a composition functor, and natural transformations that implement associativity and unitality of the composition. The morphisms of D 1 are called 2-morphisms. The horizontal 2-category of a double category D is the 2-category consisting of the objects of D 0 , horizontal 1-morphisms and 2-morphisms between parallel 1-morphisms.
We continue with a sketch of the double category of bordisms with tangential structure for a chosen group homomorphism ξ : G −→ GL 2 , which we denote is loosely speaking a stratified 2-dimensional manifold in which the d-manifold is embedded. We will not need the precise definition, but we refer to A diffeomorphism of such a bordism is a diffeomorphism which is compatible with the boundary parametrisation maps. The objects of (Bor d G ) 0 are compact 0-dimensional manifolds with haloes with G-structure, and morphisms are diffeomorphisms of these haloes with Gstructure. The objects of (Bor d G ) 1 are 1-bordisms with G-structure (recall Section 2.1), and its morphisms are diffeomorphism classes of 2-bordisms with Gstructure. The composition functor is given by glueing of bordisms.
The double categories Bor d G are symmetric monoidal via the disjoint union. The 2-category of bordisms with G-structure Bord G 2,1,0 is defined to be the horizontal 2-category of Bor d G . We will use the notation Bord fr 2,1,0 , Bord or 2,1,0 , Bord r-spin 2,1,0 (3.36) for the 2-categories of framed, oriented and r-spin bordisms, respectively. By [WHS,Thm. 1.1] these categories inherit a symmetric monoidal structure from the respective double categories.

Functors from group homomorphisms
Consider the following commutative diagram of homomorphisms of topological groups: For a G-structure (P, q) on a surface Σ, the group homomorphism λ induces a G ′ -structure on Σ via the associated bundle construction: This construction is compatible with glueing of bordisms with tangential structure, and with disjoint union. Hence it gives rise to symmetric monoidal functors of double categories and of 2-categories: The group homomorphisms in (2.3)-(2.5) fit into the commutative diagram and induce symmetric monoidal functors Bord fr 2,1,0 Bord r-spin 2,1,0 Bord or 2,1,0 , where we use the notation of (3.39) and (3.40). The functor Λ assigns to a framed manifold the manifold with the trivial r-spin structure corresponding to the orientation induced by the framing, and Λ assigns to a haloed r-spin surface the haloed surface with the underlying orientation.

Fully extended r-spin TQFTs
In this section we consider 2-dimensional extended TQFTs with tangential structure and the cobordism hypothesis for r-spin structures, r ∈ Z 0 . For this we first recall the framed cobordism hypothesis, homotopy group actions on 2-categories, and their homotopy fixed points. The latter are expected to describe TQFTs with tangential structures, as is known to be the case for oriented (or equivalently: 1-spin) TQFTs. After a review of earlier results in the oriented case, we give a presentation of all r-spin bordism 2-categories in terms of fully dualisable objects, and prove the r-spin cobordism hypothesis (for 2-categories, not for (∞, 2)-categories).
Definition 3.6. Let B be a symmetric monoidal 2-category. A fully extended 2dimensional TQFT with G-structure valued in B is a symmetric monoidal functor We write Fun sm (Bord G 2,1,0 , B) for the symmetric monoidal 2-groupoid of fully extended TQFTs with G-structure and values in B.

The framed cobordism hypothesis
Denote with 2D 0 the symmetric monoidal 2-category freely generated by a single 2-dualisable object +, cf. [SP, Ps]. Our slightly ambiguous notation for the generating object in 2D 0 draws justification from the following fact: sending the object + ∈ 2D 0 to the positively framed (halo of a) point +.
The framed cobordism hypothesis classifies framed fully extended TQFTs in terms of fully dualisable objects: Theorem 3.8 ( [Ps,Thm. 8.1]). The 2-groupoid of framed fully extended TQFTs with target B is equivalent to the core of the 2-category of fully dualisable objects in B as a symmetric monoidal 2-groupoid: (3.44)

Homotopy G-actions on 2-categories
In order to state the cobordism hypothesis with orientation and more generally with r-spin structure, we will need the notion of homotopy action of a group on a 2-category, as well as its fixed points. Let G be a topological group. The homotopy action of G on a symmetric monoidal 2-category B is a monoidal functor from the fundamental 2-groupoid of G to the 2-category of symmetric monoidal autoequivalences of B. On Π 2 (G) the monoidal structure comes from the group structure on G, on Aut sm it is the composition of functors. Equivalently, a homotopy G-action on B is a functor from the delooping of Π 2 (G) to the 3-category of symmetric monoidal 2categories. For G = GL + 2 ≃ SO 2 the fundamental 2-groupoid is equivalent to the 2groupoid BZ with a single object ⋆ with automorphism group the free abelian group on a single generator Z, and only identity 2-morphisms. Below we will identify Π 2 (GL + 2 ) with the 2-groupoid BZ. To define an action ρ of GL + 2 on a 2-category we only need to specify the value of ρ on the generator −1 ∈ Z. Recall from Proposition 3.2 the Serre automorphism S : Id B fd −→ Id B fd . We define the homotopy action of GL + 2 on fully dualisable objects as follows: Similarly, the homotopy action of the r-spin group is defined through the r-th power of the Serre automorphism: (3.48) Note that this action is the GL + 2 -action (3.47) composed with the functor induced from the covering map p r : GL + 2 r −→ GL + 2 in (2.5).

Presentations of r-spin bordism 2-categories
The 2-category of homotopy fixed points B G of a homotopy action ρ as in (3.46) is defined to be the 2-category of natural transformations of functors of 3-categories where the constant functor ∆ ⋆ : BΠ 2 (G) −→ {sym. mon. 2-cat.} sends the unique object in BΠ 2 (G) to the 2-category ⋆ with a single object and only identity morphisms, see [HSV,. It is expected that B G is the 3-limit of the functor (3.46), but we are not aware of a rigorous development of the theory of 3-limits.
By the cobordism hypothesis it is expected that 2-dimensional fully extended TQFTs with G-structure and target B are classified by homotopy fixed points of a G-action on (B fd ) × , originating from the G-action on Bord fr 2,1,0 . To our knowledge there is no complete proof for arbitrary G available in the literature, but in the case of orientations this is a known theorem: Theorem 3.9 ( [He,Cor. 5.9]). The 2-groupoid of oriented fully extended TQFTs with target B is equivalent to the 2-groupoid of homotopy fixed points of the SO 2action on the core of fully dualisable objects in B, The proof in [He] of this uses the presentation of Bord or 2,1,0 from [SP], which is not in terms of 2-dualisability data. We also mention that the equivalence as stated is one of 2-groupoids, but later we will see that this can be extended to an equivalence of symmetric monoidal 2-groupoids.
Let n ∈ Z 1 . Given a symmetric monoidal 2-category B, we define a 2-category 2D n (B fd ). For n = 1 this reduces to the 2-category in [HV,Thm. 4.3] which is equivalent to the SO 2 -homotopy fixed points of (B fd ) × . Later we will consider the case n = r for r-spin TQFTs with r 2.
in B fd such that the following diagram commutes: • Composition and units of 2D n (B fd ) are induced from B fd . To keep the cases n = 1 and n = 1 separate, for a given object (α, θ) ∈ 2D n (B fd ) we write λ := θ if n = 1, and for n = r / ∈ {0, 1} we write ϑ := θ. In the following we will determine a presentation of Bord or 2,1,0 and Bord r-spin 2,1,0 in terms of 2-dualisability data. The results are collected in Theorems 3.14 and 3.17, but first we need some preparation.
Lemma 3.11. Let G be a topological group, and let ξ : G −→ GL 2 be a continuous group homomorphism.
(i) Every object in Bord G 2,1,0 is isomorphic to a disjoint union of points with trivial G-structure.
Proof. Every connected component c of the underlying manifold of an object in Bord G 2,1,0 is contractible, hence the G-structure on c is trivialisable. The mapping cylinder for a trivialisation gives an isomorphism in Bord G 2,1,0 . This proves part (i). To prove part (ii), consider the commutative diagram of group homomorphisms from which we get the induced functor Λ incl as in (3.39). Composing this with the functor in (3.43) provides a symmetric monoidal functor 2D 0 Bord fr 2,1,0 (3.54) This composition sends + ∈ 2D 0 to the haloed point with trivial G-structure, and symmetric monoidality implies that the image of + is fully dualisable. The claim of part (i) then completes the proof.
A deck transformation on an r-spin surface (P, q, Σ) is an automorphism of the r-spin structure (P, q) which permutes the elements of each fibre of the Z r -bundle q : P −→ F Σ. We also refer to an r-spin bordism as a deck transformation if it is a mapping cylinder of a deck transformation. The 1-morphism components S p for p ∈ Bord r-spin 2,1,0 of the Serre functor on Bord r-spin 2,1,0 are isomorphic to deck transformations ([DSPS, Rem. 1.3.1]): For later use we recall from Example 2.3(ii) the relation between deck transformations and Nakayama automorphisms: Lemma 3.12. The Nakayama automorphisms N a : C a −→ C a of the closed Λ r -Frobenius algebra in Bord r-spin 2,1,0 (∅, ∅) are deck transformations.
Another way to express this relation is as follows: (3.56) Proof. In Bord or 2,1,0 the 1-morphism components of the Serre automorphism S are diffeomorphic to the identity, and mapping cylinders of these diffeomorphisms assemble into the modification λ. In Bord r-spin 2,1,0 the r-th power of the 1-morphism components of S are diffeomorphic to the r-th power of a deck transformation, which in turn is isomorphic to the identity, thus providing ϑ.
This motivates the following definition of a symmetric monoidal 2-category 2D n via generators and relations, for every n ∈ Z 1 . The generators of 2D n are the objects, 1-and 2-morphisms of 2D 0 (cf. Section 3.3.1) together with additional 2-morphisms for all α ∈ 2D n . The relations of 2D n are • the relations of 2D 0 , • the commutativity of the diagram for all α, α ′ ∈ 2D 0 and X ∈ 2D 0 (α, α ′ ).
We note that the condition in (3.58) expresses the naturality of S. Furthermore the θ a are components of an invertible modification θ : S n −→ 1 Id 2D n . For n = 1 we write λ α := θ α , and for n = r we write ϑ α := θ α . Theorems 3.9 and 3.10 together with the 3-categorical Yoneda lemma [Bu, Thm. 2.12] imply: Theorem 3.14. There is an equivalence of symmetric monoidal 2-categories Proof. We have a chain of equivalences all natural in B. The first equivalence is from Theorem 3.9, the second is from Theorem 3.10. To explain the last equivalence, we will define functors and then show that F −1 is in fact a weak inverse of F . The functor F is the evaluation on the generating object + ∈ 2D 1 and the corresponding generating 2-morphism λ + . In detail:

62)
• for a modification ϕ : f −→ f ′ we set F (ϕ) := (ϕ + : f + −→ f ′ + ). We need to show that F indeed lands in 2D 1 ((B fd ) × ), i. e. the diagram (3.51) commutes for X = f + and n = 1. The 2-morphism component of f for where we used monoidality of Y to obtain Y (S α ) ∼ = S Y (α) , etc. The key observation is that by functoriality and monoidality of f we have f Sα = S fα . Naturality of f implies that (3.51) then indeed commutes: Now we construct the functor F −1 . Since 2D 1 is defined in terms of generators and relations, in order to define a symmetric monoidal functor Y : 2D 1 −→ B it is enough to specify the value of Y on the generating objects and morphisms. Then one needs to check that relations in 2D 1 are sent to relations in B. The same holds for defining natural transformations and modifications, where we only need to specify their components on generators.
We need to check that F −1 is well-defined. First we note that Y = F −1 (α, θ) is indeed a functor 2D 1 −→ B, because the relations in 2D 0 are satisfied by definition of Y , and so are the relations on λ and its inverse. By symmetric monoidality, Y sends S + in 2D 1 to S α in 2D 1 ((B fd ) × ), hence naturality is satisfied as well. F −1 (X) is a natural transformation, since its components satisfy (3.64). For modifications there are no further conditions to check. By construction we have F • F −1 = Id. Moreover, we also have F −1 • F ∼ = Id, since two functors out of 2D 1 are isomorphic if they agree on generators ( [SP,Thm. 2.78]). This shows that F is an equivalence.
Finally we observe that the functor ι 1 respects the symmetric monoidal structures on 2D 1 and Bord or 2,1,0 and hence it can canonically be promoted to a symmetric monoidal functor. Thus the claim follows from the 3-categorical Yoneda lemma of [Bu, Thm. 2.12].
Remark 3.15. Using the symmetric monoidal equivalence ι 1 one also obtains a symmetric monoidal structure on the equivalence in Theorem 3.9.
where the functors K and K by definition act as the identity on objects as well as on 1-and 2-morphism generators of 2D 0 , while on the other generators we have By Theorems 3.7 and 3.14, the functors ι 0 and ι 1 are equivalences. In Section 3.3.4 below we will prove: Theorem 3.17. The functor in (3.65) is an equivalence for all r ∈ Z 1 , ι r : 2D r ∼ = −→ Bord r-spin 2,1,0 . (3.68) We also have the analogous statement of Theorem 3.10, which follows immediately by applying [HV,Thm. 4.3] to the natural transformation S r : Lemma 3.18. The homotopy fixed points of the r-spin action on B fd are B fd Spin r 2 ∼ = 2D r (B fd ) . The first equivalence is from Theorem 3.17, the last equivalence is Lemma 3.18, and the proof of the second equivalence is completely analogous to the n = 1 case in the proof of Theorem 3.14.
Remark 3.20. The proof of the oriented cobordism hypothesis in [He] (Theorem 3.9) uses the presentation of Bord or 2,1,0 of [SP], which is not in terms of 2-dualisability data, and a direct computation of the SO 2 -homotopy fixed points [(B fd ) × ] SO 2 (Theorem 3.10). In order to prove the r-spin cobordism hypothesis (Theorem 3.19) we need a presentation of Bord r-spin 2,1,0 (Theorem 3.17) in terms of 2-dualisability data, and a direct computation of Spin r 2 -homotopy fixed points [(B fd ) × ] Spin r 2 (Lemma 3.18).
Remark 3.22. Let us assume that the adjoints of 1-morphisms in B satisfy X † = † X, which is for example the case when B is pivotal. Then by the definition of the Serre automorphism (3.23) and its inverse (3.24), we have S = S −1 . Hence under this assumption, r-spin TQFTs with target B factorise through oriented TQFTs (r odd), or through 2-spin TQFTs (r even).

Proof of the r-spin cobordism hypothesis
Here we prove Theorem 3.17. To do this, we will check the conditions listed in the following Whitehead-type theorem for the functor ι r in (3.65). Lemma 3.24. The functor ι r is essentially surjective on objects.
Proof. This follows from Lemma 3.11(i).
Proof. For every connected component c of a 1-morphism P −→ F X in Bord r-spin 2,1,0 , we obtain an element δ(c) ∈ Z r as follows. If the 1-manifold of which c is a halo is closed, setc := c, otherwise letc be the r-spin surface with closed embedded 1-manifold defined by identifying the two boundary points of the embedded 1-manifold in c via the boundary parametrisation maps. This identification is possible by choosing a trivialisation of the r-spin structures of the objects parametrising the boundary of c. Consider a curve Γ : S 1 −→c parametrising the 1-manifold inc and its lift Γ : S 1 −→ Fc to the frame bundle defined by picking at every point a tangent vector to Γ and another vector so that the induced orientation agrees with the orientation underlying the r-spin structure of c. This lift is unique up to homotopy. There is a unique lift Γ : S 1 −→ P |c of Γ after fixing it at one point, as the fibres are discrete. We define δ(c) ∈ Z r to be the holonomy of Γ, which only depends on c; for more details of this construction we refer to [RW] or [RS,Sect. 5.2].
Recall that by (3.55), S + is isomorphic to a deck transformation with holonomy −1 ∈ Z r . If c =c, then pre-or post-composed with one of the adjunction 1-morphisms of +, for Lemma 3.26. The functor ι r is full on 2-morphisms.
The strategy of our proof is as follows. We describe the r-spin structure on Σ up to diffeomorphisms of r-spin surfaces with underlying diffeomorphism of surfaces the identity, which we refer to as isomorphisms of r-spin structures. Then we consider a decomposition of the oriented surface Λ(Σ) suitable for our description of r-spin structures. Finally we lift the oriented generators to r-spin generators and restore the r-spin structure up to isomorphism in the above sense. Therefore the r-spin surface we build from the generators is in particular diffeomorphic to Σ, thus representing the same 2-morphism in Bord r-spin 2,1,0 .
Step 1: Following [RW], we describe the r-spin structure of Σ in terms of holonomies along curves in the underlying oriented surface Λ(Σ) in the Z r -bundle q : P −→ F Λ(Σ).
Step 1.2.1 (definition of Σ): We define the new oriented surface Σ using the boundary parametrisation maps. Let ∂ i , i ∈ {1, . . . , |π 0 (∂Σ)|}, denote the parametrised boundary components of Λ(Σ), which may be circles or intervals. We arbitrarily single out the component ∂ 1 , and we choose a connected subset U j of an open neighbourhood of ∂ 1 for each remaining boundary component ∂ j , j ∈ {2, . . . , |π 0 (∂Σ)|} so that the U j are pairwise disjoint. Furthermore we choose a connected subset V j of an open neighbourhood of ∂ j in each remaining boundary component (j = 1). We illustrate such choices in Figure 3.3.
Using the boundary parametrisation maps we glue U j ∩ ∂ 1 to V j ∩ ∂ j . Finally we retract each remaining boundary component to a single point. The surface Σ obtained in this way is a closed surface, whose genus is the sum of g and the number of closed parametrised boundary components of Λ(Σ), with a point p i removed for each parametrised boundary component ∂ i ; see Figure 3.4. b) The surface Σ with all the curves a k , b k , u j , d i from Step 1.2.2. b) Only those curves on Λ(Σ) whose images in Σ form a minimal generating set of π 1 ( Σ).
Since near each U j and V j the r-spin surface is trivial, we obtain an r-spin structure on Σ. Also note that the set of r-spin structures on Λ(Σ) with prescribed r-spin structure near the boundary and the set of r-spin structures on Σ with prescribed r-spin structure near its punctures are in bijection by construction. These sets are in bijection with the set (3.74) where x i is the holonomy along d i and y j is the holonomy along u j , which are fixed by the boundary parametrisation.
In order to describe the r-spin structure on Σ it is enough to remember the holonomies along a set of curves that generate π 1 ( Σ). Therefore we reduce the set of curves a k , b k , d i , u j by discarding the curves u j with ∂ j ∼ = S 1 . The remaining curves are illustrated in Figure 3.5 b).
Step 2: We pick a decomposition of Λ(Σ) into oriented generators, so that • for each handle we have the decomposition as in Figure 3.6 a), • for each boundary component we have the decomposition as in Figure 3.6 b).
Note that we can require that 1-morphism components of the Serre automorphism only appear at the boundary of generating 2-morphisms in Bord or 2,1,0 and not in their interior, as in Bord or 2,1,0 there is a trivialisation of the Serre automorphism, cf. Lemma 3.13. We furthermore require that the curves a k , b k , u j , d i cross the generating 2-morphisms as in Figure 3.7.
Step 3: Recall from Lemma 3.12 that the Nakayama automorphisms N a : C a −→ C a are deck transformations. We lift the oriented generators to r-spin generators by inserting (i. e. by replacing a neighbourhood of a k and d j with) at the intersection of a k and b k , see Figure 3.7 a), Figure 3.7: Insertion of the Nakayama automorphism at intersection of curves.
The r-spin structure given by this construction has the same holonomies along the above mentioned curves as the r-spin structure of Σ. Note that a full circle along a positively oriented simple closed loop, where no insertions of N δ(a k ) appear, contributes +1 to the holonomy. Hence the two r-spin structures are isomorphic, and thus the two r-spin surfaces represent the same 2-morphisms.
Proof. Let σ, σ ′ ∈ 2D r (α, β)(X, Y ). Assume that ι r (σ) = ι r (σ ′ ), and that these are connected bordisms. By Proposition 3.16 we have and analogously for σ ′ . Hence since ι 1 is an equivalence (Theorem 3.14), we have This means that there is a sequence of relations in 2D 1 relating K(σ) and K(σ ′ ). By (3.67) the numbers of λ and λ −1 in K(σ) and K(σ ′ ) are each divisible by r. Using the coherence theorem for 2-categories, we can bundle together the relations in the sequence involving λ in tuples of r. These can be lifted to relations in 2D r via (3.67). Noting that all other relations in 2D 1 and 2D r are the same (to wit, those of 2D 0 ) and that they can hence also be lifted, it follows that σ = σ ′ .

Computing invariants of r-spin bordisms with closed boundary
With the r-spin cobordism hypothesis at hand, we can describe the closed Λ r -Frobenius algebra which classifies the non-extended r-spin TQFT associated to a fully extended r-spin TQFT. In particular we can explicitly describe the values of the non-extended r-spin TQFT on r-spin surfaces with closed boundary. For convenience we also present the corresponding results for framed TQFTs.
Recall from Sections 2 and 3.2 that Bord fr 2,1 ∼ = Bord fr 2,1,0 (∅, ∅) and Bord r-spin 2,1 ∼ = Bord r-spin 2,1,0 (∅, ∅) . (3.77) Consider the fully extended framed and r-spin TQFTs Y : Bord fr 2,1,0 −→ B and Z : Bord r-spin 2,1,0 −→ B (3.78) The corresponding non-extended TQFTs Y | : Bord fr 2,1 −→ B(½, ½) and Z| : Bord r-spin are classified by the closed Λ 0 -and Λ r -Frobenius algebras in Proposition 3.4 and Corollary 3.5, respectively. In particular, the invariants assigned to the framed and r-spin tori T (d) introduced in Section 2.3 are quantum dimensions of the circle spaces in B(½, ½): (3.80) Remark 3.28. (i) Assume that, as in Remark 3.22, the left and right adjoints of 1-morphisms in the target 2-category B agree. In this case we effectively have oriented TQFTs (r odd) with all the circle spaces being isomorphic, or 2-spin TQFTs (r even) with C α a ∼ = C α a+2 for every a ∈ Z. Accordingly, the invariants associated to framed and r-spin tori may take at most two distinct values if left and right adjoints agree in B.
(ii) For oriented and 2-spin surfaces there already exist TQFTs which compute complete invariants (the oriented TQFT of [Qu] with target Vect k computed from the relative Euler characteristic, and the 2-spin TQFT of [MS, RS] with target Vect Z 2 k computing the Arf invariant). For r > 2, TQFTs with pivotal 2-categories as targets cannot distinguish all r-spin structures, but other targets may allow for more interesting r-spin TQFTs.

Examples
By the main result of the previous section, constructing extended r-spin TQFTs amounts to finding fully dualisable objects whose Serre automorphisms are such that their r-th power is trivialisable. In Section 4.1 we increase our chances to find such objects by passing from a given target 2-category to its "equivariant completion", where we translate the condition on the Serre automorphism to a condition on the Nakayama automorphism of certain Frobenius algebras (Corollary 4.9), and we study the associated circle spaces and hence torus invariants in detail (Section 4.1.5). Then in Section 4.2 we show that every object in the 2category of Landau-Ginzburg models LG gives rise to an extended 2-spin TQFT, and we illustrate how to do computations in the equivariant completion of LG and its variants.

Equivariant completion
In this section, we consider the representation 2-category B eq of certain Frobenius algebras internal to a given symmetric monoidal pivotal 2-category B. In particular, we explicitly determine the Serre automorphisms and circle spaces associated to objects in B eq , from which invariants of extended r-spin TQFTs with values in B eq can be computed with the help of Theorem 3.23. We stress that even if the original 2-category B is pivotal, its completion B eq need not be pivotal, which in light of Remark 3.28 is a desired feature.
Throughout this section we fix a symmetric monoidal pivotal 2-category B which satisfies the condition ( * ) below. (The symmetric monoidal structure will not be relevant before Section 4.1.3.)

Equivariant completion of a 2-category
A ∆-separable Frobenius algebra on an object α ∈ B consists of (4.2) Recall, e. g. from [CR, Sect. 2.2], the notions of (bi)modules and (bi)module maps over the underlying algebra (A, µ A , η A ). If X is a right A-module and Y is a left A-module, then the relative tensor product X ⊗ A Y is the coequaliser of the canonical maps X ⊗ A ⊗ Y X ⊗ Y . Since A is a ∆-separable Frobenius algebra, the map is an idempotent. If π X,Y A splits, then X ⊗ A Y can be identified with Im(π X,Y A ), see e. g. [CR,Lem. 2.3]. Hence we will make the following assumption: ( * ) For all ∆-separable Frobenius algebras A on all objects of B, the idempotents π X,Y A split for all modules X, Y , and we choose adjunction data for A such that † A = A † as well as † µ = µ † and † ∆ = ∆ † .
Thus we have splitting maps Note that every Frobenius algebra is self-dual, so there always exist adjunction data such that † A = A † . The conditions † µ = µ † and † ∆ = ∆ † automatically hold if B is pivotal, but we do not make this stronger assumption on B.
Definition 4.1. The equivariant completion B eq of B is the 2-category whose • objects are pairs (α, A) with α ∈ B and A ∈ B(α, α) a ∆-separable Frobenius algebra; • 1-morphisms (α, A) −→ (β, B) are 1-morphisms α −→ β in B together with a B-A-bimodule structure; • 2-morphisms are bimodule maps in B; • horizontal composition is the relative tensor product, and 1 (α,A) is A with its canonical A-A-bimodule structure; • vertical composition and unit 2-morphisms are induced from B.
Equivariant completion was introduced in [CR] in connection with generalised orbifold constructions of oriented TQFTs. The attribute "equivariant" derives from the fact that an action ρ : G −→ B(α, α) of a finite group G (viewed as a discrete monoidal category G) gives rise to a ∆-separable Frobenius algebra structure on A G := g∈G ρ(g) if B(α, α) has finite direct sums, and that Gequivariantisation can be described in terms of categories of A G -modules.
The assignment B −→ B eq is a completion in the sense that (B eq ) eq ∼ = B eq , see [CR,Prop. 4.2]. Equivariant completion is the same as (unital and counital) "condensation completion" in dimension 2, as introduced in [GJF] for arbitrary dimension in the context of fully extended framed TQFTs and topological orders.

Adjoints
Recall that we assume that every ∆-separable Frobenius algebra A ∈ B(α, α) comes with chosen adjunction data such that † A = A † . Hence we can define the Nakayama automorphism and its inverse as follows: (4.5) Since we also assume † µ = µ † and † ∆ = ∆ † , the Nakayama automorphism is a map of the underlying algebra and coalgebra structures of A, and A is a symmetric Frobenius algebra iff γ A = 1 A , see e. g. [FS].
Remark 4.2. Let C be a symmetric monoidal 1-category with left duals, and let A ∈ C be a Frobenius algebra. We can endow C with right duals by setting X † := † X, ev X := ev X • b X † ,X and coev X := b X,X † • coev X using the braiding b.
In this case the inverse Nakayama automorphism of A is (4.6) Comparing this to the definition (2.10) of the Nakayama automorphisms of a closed Λ 0 -Frobenius algebra C, we see that the conventions for these two different Nakayama structures γ and N, for two different algebraic entities A and C, respectively, are not maximally aligned.
Given a B-A-bimodule X ∈ B(α, β) together with algebra automorphisms ϕ : A −→ A and ψ : B −→ B, the ψ-ϕ-twisted bimodule ψ X ϕ is given by where the unlabelled vertices on the right-hand side correspond to the original bimodule structure on X.
If the 2-category B has adjoints for 1-morphisms, then, as shown in [CR,Prop. 4.2], its equivariant completion B eq inherits this property, by twisting with Nakayama automorphisms. This implies that even if B is pivotal, B eq typically is not: Proposition 4.3. Let B be a 2-category, and let X ∈ B eq ((α, A), (β, B)) be such that the underlying 1-morphism X : α −→ β in B has left and right adjoints † X and X † , respectively. Then X also has left and right adjoints in B eq , witnessed by the adjunction maps (4.9) As a consistency check, we recall that † A † A A , † A † A A (4.10) are the canonical A-actions on † A, and the A-A-bimodule structure on A † is obtained as the mirror images of the above diagrams. These actions agree by assumption on A. From this it is straightforward to verify that are bimodule maps. Since † A = A † by assumption, it follows that (4.12) in B eq . Moreover, the special case , which is consistent with 1 (α,A) = A.

Symmetric monoidal structure
As explained in [WHS,Cor. 6.12], the equivariant completion B eq is the horizontal 2-category of a symmetric monoidal double category B eq , which satisfies the conditions under which the symmetric monoidal structure of B eq is passed on to B eq : Proposition 4.4. B eq has a symmetric monoidal structure induced from B.
Here we collect the ingredients of the symmetric monoidal structure on B eq in graphical presentation, using the conventions of Section 3.1. The monoidal product on objects (α, A), (α ′ , A ′ ) ∈ B eq is given by where the ∆-separable Frobenius structure on A A ′ ≡ A B A ′ is as follows: (4.14) It follows that the Nakayama automorphism of A A ′ factorises with respect to , On 1-morphisms X ∈ B eq ((α, A), (β, B)) and X ′ ∈ B eq ((α ′ , A ′ ), (β ′ , B ′ )), the monoidal product is X B X ′ with the left (A A ′ )-and right (B B ′ )-action induced from B. On 2-morphisms, we have Beq = B . From now on we will denote the monoidal product of both B and B eq simply as .

Duality and Serre automorphism
If α ∈ B is dualisable with duality data (α # , ev α , coev α ), then every object (α, A) ∈ B eq is dualisable with where we used the isomorphism induced by the inverse cusp isomorphism c −1 l in (3.18), and similarly with coev (α,A) and c −1 r . These isomorphisms together with µ A also give the above adjunction morphisms their bimodule structures.
The Frobenius algebra structure ( up to cusp isomorphisms as needed. We illustrate this with the multiplication where the last expression is shorthand for the defining Gray diagram in the middle. Note that there is a canonical isomorphism A ∼ = A ## in B (see [BMS,Fig. 32]), which we leave implicit. Similarly, we denote the other structure maps of A # as (4.22) The enveloping algebra of A is (4.23) The cusp isomorphisms together with µ A give a canonical right A e -module structure on ev (α,A) and a left (A e ) # -module structure on coev (α,A) .
Lemma 4.6. If α ∈ B is dualisable, then for (α, A) ∈ B eq we have that up to cusp isomorphisms in B.
Proof. Up to cusp isomorphisms, ev A # : (4.25) and similarly for the coevaluations: (4.26) Hence together with (4.21) and (4.22), we find (4.27) We now turn to full dualisability, which is another property that is compatible with equivariant completion: Proposition 4.7. Let α ∈ B be fully dualisable. Then every (α, A) ∈ B eq is fully dualisable.
In particular, according to (4.8) we have (4.28) Hence the Serre automorphism of (α, A) ∈ B eq is (4.29) Applying the 2-isomorphisms b A,A and b −1 A † ,1α , the inner A-line can be moved to the right and the A † -line can be moved to the left, respectively. Alternatively, the A † -line can be moved to the left of the b α,α -line with the help of two cusp isomorphisms, and then to the right by b A † ,1α . Thus we have shown: Proposition 4.8. Let α ∈ B fd . Then (α, A) ∈ B fd eq , and (4.30) Below we will frequently not display A = 1 (α,A) and simply write S (α, Corollary 4.9. Let r ∈ Z 1 , α ∈ B fd , and (α, A) ∈ B eq such that S r α ∼ = 1 α and γ r A = 1 A in B. Then there is an r-spin TQFT Z : Bord r-spin 2,1,0 −→ B eq + −→ (α, A) . (4.31) Proof. Combining (4.30) with A † ∼ = A γ −1 A , Lemma 3.18 and Theorem 3.19, we see that any choice of isomorphism S r α ∼ = 1 α determines a Spin r 2 -homotopy fixed point in (B fd ) × .

A Frobenius algebra
For (α, A) ∈ B fd eq and a ∈ Z, the a-th circle space (recall (3.29)) is (4.32) By Proposition 4.3 we have (4.33) and by Proposition 4.8 together with (4.11) and † A = A † we have Our next goal is to explicitly describe the closed Λ 0 -Frobenius structure on the circle spaces C (α,A) a in B eq . This means that we will determine the (co)multiplication and (co)unit of (3.31)-(3.34) of C (α,A) a directly in terms of data in B. In doing so we will frequently use the fact that the 2-morphism (4.35) in B induces the identity S x (α,A) ⊗ A S y (α,A) −→ S x+y (α,A) , up to the isomorphism in (4.34). From now on we will no longer display all A-lines in diagrams that represent 2-morphisms in B eq , since A = 1 (α,A) . Accordingly, we abbreviate (4.35) and its inverse as (4.36) With the above preparations, we can present the isomorphism (3.35) in the case of the equivariant completion: Lemma 4.10. Let α ∈ B fd . Then for any (α, A) ∈ B eq , there are mutually inverse isomorphisms given by: Proof. Repeated use of (4.11) together with standard manipulations of string diagrams for ∆-separable Frobenius algebras shows that f A , f ′ A are indeed bimodule maps, and that f A • f ′ A = 1 † A # . Since their source and target are isomorphic, it follows that f A and f ′ A indeed represent mutually inverse 2-morphisms in B eq . We have now expressed all the ingredients of the closed Λ 0 -Frobenius structure on {C (α,A) a } in B eq directly in terms of data in B. With this the Nakayama automorphisms N (α,A) a can be computed: Proposition 4.11. Let α ∈ B fd . Then for any (α, A) ∈ B eq , we have (4.39) Proof. Our task is to compute N (α,A) a as defined in (2.10) in B eq . We will first compute ε −1 • µ a,−a and ∆ a,−a • η 1 in B eq , starting with ε −1 • µ a,−a . Using the notation introduced in (4.36), we have and inserting the expression for f A in (4.38) into (3.32), we have (4.41) In the composition ε −1 • µ a,−a , we first use where here and below we suppress S α -strands. This expression cancels with another subdiagram of ε −1 • µ a,−a , leaving (4.43) The second step uses the properties of ∆-separable Frobenius algebras and (4.5); moreover, here and below we suppress ev α and its adjoints (as they are only spectators in our string diagram manipulations). Analogously, we arrive at with (where we continue to suppress ev α ) Repeatedly using the defining properties of ∆-separable Frobenius algebras as well the properties of the Nakayama automorphism γ A # collected in Section 4.1.2, a straightforward but lengthy computation shows that the above string diagram is equal to (4.48) Putting ev α , ev † α , S 1−a α back in, a final application of the isomorphisms (4.12) and (4.34) allows us to identify (4.48) with (4.49) Post-composing with (4.46) thus completes the proof.
Combining Proposition 4.11 with the isomorphisms (4.30) and (4.34), we obtain closed Λ r -Frobenius algebras from ∆-separable Frobenius algebras A on fully dualisable objects if the r-th power of the Nakayama automorphism of A is the identity: Corollary 4.12. If for r ∈ Z 1 there is an isomorphism S r α ∼ = 1 α in B fd , and for (α, A) ∈ B eq we have γ r A = 1 A , then there is an induced closed Λ r -Frobenius algebra structure on {C can be simplified. For ease of presentation, we further assume that S α ∼ = 1 α ; in Section 4.2 below we will see how this restriction can be lifted in practice.
Lemma 4.13. Let α ∈ B fd with S α ∼ = 1 α , and assume that for (α, A) ∈ B eq we have C ). Then and N (α,A) a corresponds to post-composition with γ A .
Note that the above result further elucidates the relation between the two different notions of "Nakayama morphism" N and γ.
Proof of Lemma 4.13. We have (4.51) In the second step we used (4.32) and adjunction for ev (α,A) ; the third step is the isomorphism the fourth step is (4.34) together with the assumption S α ∼ = 1 α ; the fifth step is a standard computation with ∆-separable Frobenius algebras along the lines of [BCP,Sect. 3.2].

Landau-Ginzburg models
In this section we briefly review the 2-category of Landau-Ginzburg models LG and note that every object in LG gives rise to an extended 2-spin TQFT. Then we apply the results of Section 4.1 to a closely related 2-category LG •/2 and consider the simplest non-trivial example.
Recall from [CM1] that for every fixed field k, there is a 2-category LG whose objects are pairs (k[x 1 , . . . , x n ], W ), where n ∈ Z 0 and W = 0 ∈ k if n = 0, while for n > 0, W ∈ k[x] ≡ k[x 1 , . . . , x n ] is such that the Jacobi algebra is finite-dimensional over k. We refer to such polynomials W as potentials. The Hom categories of LG are idempotent completions of homotopy categories of finite-rank matrix factorisations. Hence up to technicalities with idempotents (which will not be relevant to our discussions below), a 1-morphism The Hom sets of 2-morphisms (X, d X ) −→ (X ′ , d X ′ ) consist of the even cohomology classes of the differential defined on Z 2 -homogeneous maps as , and the unit 1morphism of (k[x 1 , . . . , x n ], W ) is 1 W = (I W , d I W ) with (4.55) where {θ i } is a chosen k[x ′ , x]-basis of k[x, x ′ ] ⊕n , and (4.56) A straightforward computation shows that End(1 W ) ∼ = Jac W in LG. Every 1-morphism X ≡ (X, d X ) ∈ LG((k[x 1 , . . . , x n ], W ), (k[z 1 , . . . , z m ], V )) has a left adjoint † X and a right adjoint X † , and † X ∼ = X † iff m = n mod 2. The associated adjunction 2-morphisms are explicitly known, see [CM1,Thm. 6.11], or [CM2] for a concise review.
The monoidal 2-category LG has a symmetric braiding, whose 1-morphism components b V,W are given by 1 V +W (up to a reordering of variables), while the 2-morphism components are compositions of canonical module isomorphisms and structure maps of the underlying 2-category LG. For details we refer to [CMM,Sect. 2.3]. In summary, we have: Theorem 4.14 ( [CM1,CMM]). For every field k, the 2-category of Landau-Ginzburg models LG has a symmetric monoidal structure such that LG = LG fd .

Remark 4.15. A variant of
LG is the symmetric monoidal 2-category LG •/2 , which is defined analogously to LG, but 2-morphisms are given by both even and odd cohomology of the differentials δ X,X ′ in (4.54), but with classes −ζ and +ζ identified. This ad hoc Z 2 -quotient allows to stay within the realm of 2-categories (as opposed to super 2-categories) while allowing odd 2-morphisms, compare [KR] and [CMM,Rem. 3.11(ii)].
Theorem 4.14 also holds for LG •/2 , i. e. LG •/2 = (LG •/2 ) fd . where [n] denotes the n-fold application of the shift functor [1], which sends a matrix factorisation (X 0 ⊕ X 1 , d X ) to (X 1 ⊕ X 0 , −d X ). It follows that [2] is the identity functor, and one finds that Hom(1 W , 1 W [n]) ∼ = δ n,0 mod 2 · Jac W in LG, as LG has only even cohomology classes as 2-morphisms, while Hom(1 W , 1 W [n]) ∼ = Jac W [n] is purely odd in LG •/2 if n is odd. As a consequence, (k[x 1 , . . . , x n ], W ) determines an extended oriented TQFT with values in LG iff n is even, and it determines an extended oriented TQFT with values in LG •/2 for every value of n.  [SP, He] to the duality data of an object (k[x], W ) in LG or in LG •/2 , one recovers the (non-semisimple) commutative Frobenius algebra Jac W from the circle, the pair-of-pants, and the disc.
As an immediate consequence of Theorem 3.19, Lemma 3.18, (4.57) and the isomorphism Aut(1 W ) ∼ = k, we find that every potential depending on an odd number of variables gives rise to a proper extended spin TQFT: Theorem 4.17. Every object W ≡ (k[x 1 , . . . , x n ], W ) ∈ LG gives rise to a unique-up-to-isomorphism extended 2-spin TQFT valued in LG. These TQFTs factor through the oriented bordism 2-category iff n is even.
It is straightforward to compute that C W a ≡ C (k[x 1 ,...,xn],W ) a ∼ = Jac W [n · (1 − a)] in LG(½, ½) ∼ = vect Z 2 k (4.58) for a ∈ {0, 1}. Hence these circle spaces are the zeroth Hochschild homology and cohomology, respectively, of the differential graded category of matrix factorisations, first computed in [Dy]. Moreover, for the Nakayama automorphisms we have N W a = (−1) n·(1−a) · 1 C W a . We now turn to the equivariant completion of LG •/2 to look for r-spin TQFTs that can detect more r-spin structures than oriented TQFTs. One type of exam-ple+ of ∆-separable Frobenius algebras with trivialisable r-th power of its Serre automorphism is the algebra A G mentioned in Section 4.1.1, in the case G = Z r .
The computational techniques used in Example 4.18 can analogously be applied to more involved examples. For instance, there are Z r -actions on k[x 1 , x 2 ] which leave W = x r 1 + x 2r 2 invariant, and the associated (LG •/2 ) eq -valued TQFTs may detect more than two r-spin structures on the torus. We leave such computations as well as the application of the theory developed in Section 4.1 to the 2-category of Q-graded Landau-Ginzburg models LG gr (see [CM1] or [CMM, Sect. 2.5]) to future work.